Non-local analysis of families of periodic solutions in autonomous systems

One-parameter families of periodic solutions arising from equilibrium positions of an autonomous system are considered. It is shown that they may be divided into families of the first and second kind; families of one kind cannot be identical when continued as the parameter is varied. As a result, a lower bound is obtained for the number of families that may be continued to arbitrary large values of the norm or the period, and an estimate is also obtained for the number of periodic solutions with a given minimal period. Additional properties of these families are established for Hamiltonian systems satisfying certain symmetry conditions. The results are illustrated for an articulated pendulum. (C) 2000 Elsevier Science Ltd. All rights reserved.